IntroductionModeling with DecimalsIrrational NumbersConclusionIntroduction Modeling with Decimals Irrational Numbers ConclusionMATH 113Section 5.4: Decimals, Exponents, and RealNumbersProf. Jonathan DuncanWalla Walla UniversityWinter Quarter, 2008Introduction Modeling with Decimals Irrational Numbers ConclusionOutline1Introduction2Modeling with Decimals3Irrational Numbers4ConclusionIntroduction Modeling with Decimals Irrational Numbers ConclusionDecimalsDecimals are an even more recent invention than fractions.History of DecimalsIn 1585 Flemish mathematician Simon Stevinpublished a small pamphlet, La Thiende (“TheTenth”), in which he presented an account ofdecimal fractions and their daily use. Though hedid not invent decimal fractions and hisnotation was clumsy, he established the use ofdecimals in day-to-day mathematics.Decimal NotationTo this day decimal notation varies.United States: 3.14United Kingdom: 3·14Continental Europe: 3,14Introduction Modeling with Decimals Irrational Numbers ConclusionDecimals and Place ValueIn order to tie decimals into our existing number system, we willstart with an e xamination of place values.Decimal Place ValuesIn base 10, decimals have place values which are negative powersof 10, or positive powers of110..10210110010−110−2ExampleSuppose we used a base 5 system. What would the “decimal”place values look like?Introduction Modeling with Decimals Irrational Numbers ConclusionDecimals and FractionsAs decimals were originally intended to simplify working withfractions, there is a strong connection between the two concepts.Decimals and FractionsDecimals are an alternative to fractions with denominators whichare powers of 10. For example, 0.3 = 0 × 1 + 3 ×110=310.ExampleConvert each fraction to a decimal:121411613ExampleConvert each decimal to a fraction:0.3 0.045 0.23Introduction Modeling with Decimals Irrational Numbers ConclusionThe Importance of 0Zero plays an important role in our numbering system. This role isespecially important when writing decimal numbers.The Role of 0Zero (0) acts as a place holder in our numeration system. Indecimal numbers, leading zeros can be especially im portant.Cases where Zeros Make A DifferenceNote the following:203 6= 23 .203 6= .23 .023 6= .23Cases where Zeros are OptionalNote the following:0.23 = .23 .230 = .23Introduction Modeling with Decimals Irrational Numbers ConclusionBase 10 BlocksBase 10 blocks can be used to help model decim als as long as weare sure to designate the unit.ExampleModel the following decimals using base 10 blocks.10.2920.03230.30Sometimes models, converting to fractions, or padding w ith zeroscan help us compare decimals.ExampleArrange the three decimals from the previous example indescending order.Introduction Modeling with Decimals Irrational Numbers ConclusionModeling Decimal Ad ditionModeling decimal addition is relatively straight forward. Once cansimply use base 10 blocks and the set model of addition.ExampleUse base 10 blocks to model and find 6.5 + 1.25.Models can also help to justify certain important practices in ourstandard addition algorithm.ExampleUsing blocks, explain why we need to line up the decimal pointbefore we add two decimal numbers together.Introduction Modeling with Decimals Irrational Numbers ConclusionModeling Decimal Mul tiplic ationThe area model works well for showing decimal multiplication.Again, we must be sure to designate a unit and we must alsodifferentiate between linear units and square units.ExampleUse an area model to find 3.4 × 2.3.ExampleNow perform the multiplication 3.4 × 2.3 using the standardmultiplication algorithm. Can you justify this algorithm usingblocks? In particular, why do we count up the number of decimalplaces in the factors to place the decimal in the product?Introduction Modeling with Decimals Irrational Numbers ConclusionModeling Decimal D ivis ionJust as with fractions, the repeated subtraction model can help ussee how to divide decimals.ExampleUse base 10 blocks and repeated subtraction to find 4.5 ÷ 2.1. Becareful in expressing any remainder.ExampleNow perform the division 4.5 ÷ 2.1 using the standard divisionalgorithm. Can you justify this algorithm using blocks? Inparticular, why can we move the decimals in the divisor anddividend?Introduction Modeling with Decimals Irrational Numbers ConclusionA New Operation – ExponentiationAlthough we have been using the concept of an exponent to findplace values, we have yet to formally introduce this operation.Multiplication as Repeated AdditionRecall that the product a × b can be thought of as:a × b = b + b + ··· + b| {z }a timesExponentiationFor any number b and an integer a, the expression bameans thatwe multiply a factors of b together. a is called the exponent and bis called the base in this expression. Symbolically:ba= b × b ×··· × b|{z }a timesIntroduction Modeling with Decimals Irrational Numbers ConclusionExponentiation and Scien tific N otationExampleFind the value of 2afor a = 4, 3, 2, 1, 0, −1, and −2.Exponentiation is used together with decimals to write numbers ina form called Scientific Notation.Scientific NotationIn scientific notation numbers are written as d × 10awhere d is adecimal number and a is any integer.ExampleConvert between scientific and standard notation as appropriate.13, 425, 47120.0000000014641.25 × 10652.13 × 10−7Introduction Modeling with Decimals Irrational Numbers ConclusionAre Decimals and Fractions Different?As we saw at the beginning of the class, decimals were intended tomake working with fractions easier. However, are decimals andfractions really the same thing?ExampleCan every fraction be written as a decimal? If so, what types ofdecimals? If not, why not?ExampleCan every decimal be written as a fraction? If so, justify youranswer. If not, give an e xample.Introduction Modeling with Decimals Irrational Numbers ConclusionProof that√2 is IrrationalNumbers which can not be written as fractions include π and√2.ExampleShow that√2 is not a rational number.If it were, then there are integers a, b with√2 =abwhere a and bhave no common factors.Multiplying by b,√2b = a.Squaring, 2b2= a2.We see from the above t hat a2is even. Therefore, a is also even.If a is even, then a2is a multiple of 4, say a2= 4k .Simplifying, b2= 2kAs before, b2must be even, so b is also.But then a and b are both even, so they have a common factor of 2.Since this can’t happen, there are no a, b with√2 =ab.Introduction Modeling